WATER RESOURCES RESEARCH, VOL. 40, W06301, doi:10.1029/2003WR002496, 2004
A mechanistic model for river incision into bedrock
by saltating bed load
Leonard S. Sklar
Department of Geosciences, San Francisco State University, San Francisco, California, USA
William E. Dietrich
Department of Earth and Planetary Science, University of California, Berkeley, California, USA
Received 16 July 2003; revised 20 March 2004; accepted 6 April 2004; published 18 June 2004.
[1] Abrasion by bed load is a ubiquitous and sometimes dominant erosional mechanism
for fluvial incision into bedrock. Here we develop a model for bedrock abrasion by
saltating bed load wherein the wear rate depends linearly on the flux of impact kinetic
energy normal to the bed and on the fraction of the bed that is not armored by transient
deposits of alluvium. We assume that the extent of alluvial bed cover depends on the ratio
of coarse sediment supply to bed load transport capacity. Particle impact velocity and
impact frequency depend on saltation trajectories, which can be predicted using empirical
functions of excess shear stress. The model predicts a nonlinear dependence of bedrock
abrasion rate on both sediment supply and transport capacity. Maximum wear rates occur
at moderate relative supply rates due to the tradeoff between the availability of abrasive
tools and the partial alluviation of the bedrock bed. Maximum wear rates also occur at
intermediate levels of excess shear stress due to the reduction in impact frequency as grain
motion approaches the threshold of suspension. Measurements of bedrock wear in a
laboratory abrasion mill agree well with model predictions and allow calibration of the one
free model parameter, which relates rock strength to rock resistance to abrasive wear. The
model results suggest that grain size and sediment supply are fundamental controls on
bedrock incision rates, not only by bed load abrasion but also by all other mechanisms that
INDEX TERMS: 1824 Hydrology:
require bedrock to be exposed in the channel bed.
Geomorphology (1625); 1815 Hydrology: Erosion and sedimentation; 1886 Hydrology: Weathering (1625);
1899 Hydrology: General or miscellaneous; KEYWORDS: abrasion, bedrock incision, erosion, landscape
evolution, saltation
Citation: Sklar, L. S., and W. E. Dietrich (2004), A mechanistic model for river incision into bedrock by saltating bed load, Water
Resour. Res., 40, W06301, doi:10.1029/2003WR002496.
1. Introduction
[2] Surprisingly little is known about the physical controls on rates of erosion of bedrock river beds [Whipple et
al., 2000a], given the importance of river incision into
bedrock in driving landscape evolution [e.g., Howard et
al., 1994] and linking climate and tectonics to topography
[e.g., Molnar and England, 1990]. Most numerical models
of landscape evolution use bedrock incision rules that are
related to the physics of bedrock wear only through simple
scaling arguments, typically by assuming that incision rate
is proportional to some measure of flow intensity such as
unit stream power [Seidl and Dietrich, 1992; Anderson,
1994; Tucker and Slingerland, 1994; Willett, 1999] or
average boundary shear stress [Howard and Kerby, 1983;
Howard, 1994; Moglen and Bras, 1995; Tucker and
Slingerland, 1996]. These models are difficult to apply
in real landscapes because they lump the potentially
complex influence of fundamental variables such as rock
strength, channel slope, discharge, sediment supply and
Copyright 2004 by the American Geophysical Union.
0043-1397/04/2003WR002496$09.00
grain size, into a set of poorly constrained parameters
unrelated to any particular erosional mechanism. Model
parameters are calibrated using longitudinal profiles of
rivers with a known incision history [e.g., Seidl et al.,
1994; Stock and Montgomery, 1999; Whipple et al.,
2000b; van der Beek and Bishop, 2003] or by assuming
steady state topography [e.g., Sklar and Dietrich, 1998;
Snyder et al., 2000], in effect using landscape form to infer
process. Here we take the reverse approach, and develop a
physically based model for bedrock incision by a single
mechanism, abrasion by saltating bed load. Our goals are
(1) to explore whether detailed consideration of the
mechanics of bedrock wear will lead to a view of river
incision fundamentally different from the simple scaling
models, (2) to use the model to refine mechanistic hypotheses that can be tested in the field and in laboratory
experiments, and (3) to explain river profile form in terms
of the dynamics of bedrock incision.
[3] Many erosional mechanisms operate at the interface
between a river and its rock bed, including cavitation,
dissolution, abrasion by both bed load and suspended load,
and the plucking of rock fragments by fluid shear stresses
acting without the aid of sediment impacts. We focus on bed
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Figure 1. Photographs of two incising rivers with contrasting alluvial cover. (a) Smith River, Oregon
coast range mountains. (b) South Fork Eel River, northern California coast range mountains. See color
version of this figure at back of this issue.
load abrasion for several reasons: (1) all rivers carry
sediment in some quantity so abrasion should be a ubiquitous process; (2) other mechanisms occur only under limited
conditions, for example cavitation requires extreme flow
velocities and plucking may primarily occur in weak or
highly fractured rock; (3) particle impacts are an efficient
pathway for the transfer of energy and momentum from the
flow to the bed; (4) by definition, bed load interacts with the
underlying bedrock far more frequently than does suspended load; (5) much is known about the motion of
saltating bed load from both theoretical and experimental
studies; and (6) we have direct experimental evidence that
saltating bed load causes measurable bedrock wear [Sklar
and Dietrich, 2001]. Furthermore, by focusing on abrasion
by saltating bed load, we are forced to consider in detail the
role of sediment in influencing incision rates, not only in
providing tools for abrasion, but also in controlling the
extent of bedrock exposure on the channel bed. Ultimately,
by focusing on bed load abrasion, we hope to gain insight
into the potential feedbacks between incision rate, sediment
supply and grain size that may strongly influence landscape
response to changes in climate or rock uplift rate.
[4] Actively incising channels in hilly and mountainous
terrain exhibit a wide range in the extent of alluvial cover
and size distribution of bed sediment. Figure 1 shows
photographs of two bedrock canyon channel beds in the
Coast Range mountains of southern Oregon and northern
California. The bedrock bed of the Smith River is nearly
devoid of sediment deposits, in contrast to the South Fork
Eel River, where bedrock is exposed only in isolated patches
amid a nearly continuous mantle of coarse alluvium. Yet
both of these channels, which are separated by about
150 km, are actively incising through similar marine
sedimentary rocks. To what extent does the size and rate
of sediment supply to these two channels contribute to or
inhibit the lowering of the bedrock bed?
[5] Recent experimental results from bedrock abrasion
mills [Sklar and Dietrich, 2001] support the hypothesis of
Gilbert [1877] that the quantity of sediment supplied to the
river should influence bedrock incision rates in two essential yet opposing ways: by providing tools for abrasion of
exposed bedrock and by limiting the extent of exposure of
bedrock in the channel bed. Incision is limited at lower
supply rates by a shortage of abrasive tools (what we term
the ‘‘tools effect’’) and at higher supply rates by partial
burial of the bedrock substrate beneath transient sediment
deposits (‘‘the cover effect’’). Thus, as observed experimentally, the maximum rate of river incision should occur when
the stream receives a moderate supply of sediment, relative
to its sediment transport capacity. It follows that the size
distribution of sediment grains supplied to the channel
should also influence incision rates because only the coarser
fraction is capable of forming an alluvial cover and because
the finer fraction is carried in suspension and rarely collides
with the bedrock bed. As suggested by the bedrock abrasion
mill results [Sklar and Dietrich, 2001], the most efficient
abrasive tools are sediments of intermediate size, large
enough to travel as bed load rather than suspended load
but not so large as to be immobile.
[6] Gilbert’s hypothesis is also consistent with numerous
field observations. Sediment clearly contributes to incision
through the scouring of potholes [e.g., Alexander, 1932] and
longitudinal grooves [Wohl, 1992a]. Boulders deposited in
extreme flows [Wohl, 1992b] or eroded from steep canyon
walls [Seidl et al., 1994] have been shown to shut off the
incision process over decadal to millennial timescales. In
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contrast, incision under a thin alluvial cover has been
documented in a badlands eroded into weak shale [Howard
and Kerby, 1983]. Bedrock channel beds partially mantled
with alluvial deposits are common in actively incising
terrain [Howard, 1998] and occur even in lowland coastal
streams [Ashley et al., 1988]. Montgomery et al. [1996] and
Massong and Montgomery [2000] have shown that, for a
given drainage area, a wide range in extent of alluviation
can occur over a narrow range in channel slope, due in part
to the influence of large woody debris. This suggests that
the extent of bedrock exposure in incising channels may be
highly dynamic.
[7] Sediment supply and grain size must be explicitly
represented in bedrock incision models to adequately capture the potentially complex influence of sediment on
incision rates. Although to first approximation the longterm flux of sediment to a given channel reach will be
simply the product of the landscape denudation rate times
the upstream drainage area, the supply rate of sediment in
the bed load size class will be considerably less. Some mass
eroded from upstream is transported as dissolved load. How
much of the solid load moves as bed load, and thus
contributes to bed load abrasion and partial bed cover, will
depend on the size distribution of sediment entering the
channel network, the rate of particle comminution with
downstream travel distance, and the fluid shear stress
available to transport sediment. Unlike fully alluvial channels in depositional environments, the sediment load in
mixed bedrock-alluvial channels is commonly assumed to
be less than the sediment transport capacity, because there is
a limited supply of sediment available to be scoured from
the channel bed. Therefore the transport capacity of adjacent
upstream reaches cannot be used as a proxy for local
sediment supply. At the short timescale of individual
incision events, sediment delivery can be highly episodic,
particularly in steep landslide-dominated terrain [e.g.,
Benda and Dunne, 1997]. Significant spatial and temporal
variability in sediment supply should also occur due to
heterogeneous lithology, nonuniform rates of landscape
lowering, fans and other upstream sediment storage elements, and the discrete locations of sediment input such as
tributary junctions and stream bank failures.
[8] Few bedrock incision models have attempted to explicitly account for the influence of sediment supply and grain
size on incision rate, although many landscape evolution
models [e.g., Howard, 1994; Tucker and Slingerland, 1994]
represent the cover effect as a sudden shift from bedrock to
alluvial conditions when sediment supply exceeds transport
capacity. Beaumont et al. [1992] proposed that bedrock
incision could be treated like alluvial incision, with incision
rate scaling with the sediment transport capacity in excess of
the sediment supply, in effect modeling the coverage effect
but not the tools effect. Foley [1980] modified Bitter’s
[1963a, 1963b] sandblast abrasion model to apply to wear
by saltating bed load, accounting for the tools effect but
neglecting the coverage effect. A preliminary version of the
full model developed here [Sklar et al., 1996; Sklar and
Dietrich, 1998] captured both the coverage and tools effects
and has inspired efforts to incorporate sediment into the
simple stream power incision rule [Whipple and Tucker,
2002]. Slingerland et al. [1997] used studies of slurry
pipeline wear [e.g., de Bree et al., 1982] to support the view
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that the coverage and tools effects should occur in rivers. Here
we describe in detail a model for bedrock incision by saltating
bed load based directly on the mechanics of bed load sediment
transport and the wear of brittle materials by particle impacts.
[9] The paper is divided into four sections. We begin by
deriving an expression for the rate of bedrock incision by
saltating bed load, using established theory for wear of
brittle materials by low-velocity particle impacts, in which
the erosion rate depends on saltation hop length and impact
velocity. We then use data from a suite of published studies
on saltation trajectories to parameterize the model directly
in terms of shear stress. Next, we explore the sensitivity of
the predicted bedrock abrasion rates to variations in sediment supply, shear stress and grain size, and then use a
nondimensional framework to collapse the full model behavior onto a single graph. Finally, we compare the model
predictions to laboratory measurements of bedrock wear
rates [Sklar and Dietrich, 2001] to calibrate the one adjustable parameter, which characterizes rock resistance to
fluvial abrasion.
2. Model Development
2.1. Assumptions
[10] Our goal in constructing this model is to capture, in a
general way, how the rate of bedrock incision by bed load
abrasion depends on rock strength, shear stress, sediment
supply, and grain size. We do not seek to predict the
absolute magnitude of abrasion rates precisely, nor do we
attempt to describe the exact motion of any individual
sediment grain. Rather, the model is intended as a tool to
quantify Gilbert’s [1877] hypothesis and translate the experimental observations of the influence of sediment on
rates of bedrock wear [Sklar and Dietrich, 2001] from a
closed rotational abrasion mill to the open translational
setting of natural river channels. In attempting to balance
physical realism with simplicity, we seek to represent the
mechanics at a sufficient level of detail to capture the
essential elements, while minimizing the introduction of
free parameters and the use of unrealistic or untestable
assumptions.
[11] We limit our analysis to abrasion of rock by bed load,
neglecting all other possible mechanisms for fluvial incision. We assume that all bed load motion is by saltation; in
effect we assume that rolling and sliding either do not cause
significant wear or that the net effect of those processes
scales with the effect of saltating bed load. We consider
flow through a simple rectangular cross section with a
planar bed. The assumption of vertical banks of infinite
height is equivalent to a confined canyon channel without
overbank flow onto a floodplain because flow width does
not change with stage. Finally, we assume for simplicity that
all bed load is composed of spherical grains of uniform size.
[12] The model is developed at the temporal scale of a
constant discharge (i.e., on the order of several hours), and
is essentially a zero-dimensional spatial model, simulating
erosion of a unit area of channel bed. We do not attempt to
account for cross channel variations in shear stress, local
variation in rock strength, or other reach-scale spatial
heterogeneities, although these factors may influence the
rate of incision by bed load abrasion and by other mechanisms. Importantly, we assume that the net effect of spatial
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[16] Here we treat bedrock as an elastic brittle material,
and thus do not consider ductile or nonelastic behavior. The
net transfer of kinetic energy (DeN) in a collision between a
small particle and an essentially infinite rock mass is
1
DeN ¼ =2 Mp Ui2
Figure 2. Idealized rectangular channel eroded by saltating bed load impacts. Shaded areas represent transient
alluvial deposits that shield underlying bedrock from
erosive effect of bed load sediment flux (Qs).
variations in sediment supply effects can be parameterized
at the scale of the unit bed area.
2.2. General Expression
[13] Consider an idealized flux of saltating bed load,
moving across an approximately planar river bed composed
of a mixture of exposed bedrock and alluvial patches, as
depicted in Figure 2. Bedrock erosion occurs as the integrated result of numerous grain collisions with exposed
bedrock. Many nondestructive grain impacts occur as sediment travels across local sediment deposits where bedrock
is covered. Bedrock incision rate (E) can be expressed
simply as the product of three terms: the average volume
of rock detached per particle impact (Vi), the rate of particle
impacts per unit area per unit time (Ir), and the fraction of
the river bed made up of exposed bedrock (Fe)
E ¼ Vi Ir Fe :
ð1Þ
Below we develop expressions for each of these terms using
empirical relationships for wear by particle impacts, bed
load sediment transport capacity, and the trajectory of
saltating bed load motion.
[14] Saltating grains move in a characteristic trajectory
(Figure 3). The magnitude of particle velocities and saltation hop length and height are well described by functions
of boundary shear stress in excess of the threshold of
motion [e.g., Wiberg and Smith, 1985], as is bed load
sediment transport capacity [e.g., Gomez and Church,
1989]. Ultimately, we will obtain an expression for erosion
rate in terms of sediment supply rate, grain size, excess
shear stress, and rock strength.
2.3. Volume Eroded Per Unit Impact (Vi)
[15] Erosion of brittle materials by low-velocity particle
impacts occurs through the formation, growth and intersection of a network of cracks [Engle, 1978]. Cracks occur due
to tensile stresses arising from elastic deformation around
the point of impact. Detachment of an eroded fragment is
generally the result of an accumulation of intersecting
cracks due to multiple particle impacts. Erosion of brittle
materials by repeated impacts is referred to as ‘‘deformation
wear’’ [e.g., Bitter, 1963a]. Although the volume eroded by
any single impact will depend on the local fracture density,
the wear rate averaged over a large number of impacts
scales with the flux of kinetic energy transferred by the
impacting grains [Engle, 1978].
Ur2
¼
1
=2 Mp Ui2
U2
1 r2
Ui
ð2Þ
where Mp is the mass of the impacting particle, Ui and Ur
are the velocities of impact and rebound respectively, and
(1 U2i /U2r ) = f(sin a) where a is the angle of impact [Head
and Harr, 1970]. Some of the kinetic energy transferred
may contribute to deformation wear of the bedrock (DeD),
while the remainder is lost (DeL) through transformation
into other energy forms such as sound and heat (i.e., DeN =
DeD + DeL). Thus, bedrock wear depends on the energy
transferred in excess of losses, DeD = DeN DeL. This
forms the basis of the classic impact wear model of Bitter
[1963a], which we write as
=2 Mp ðUi sin aÞ2 et
ev
1
Vi ¼
ð3Þ
where Vi is the mean volume eroded per impact, ev is the
total energy required to erode a unit volume of rock, and et
is the threshold energy that must be exceeded for
detachment to occur. Erosion depends on the sine of the
impact angle because the magnitude of the peak tensile
stress varies with the normal component of the impact
velocity [Engle, 1978].
[17] Note that the tangential velocity component can
contribute to erosion by scratching or gouging, referred to
as ‘‘cutting wear’’ [Bitter, 1963b]. Cutting wear is important
in ductile materials and when the impacting particles are
highly angular, but is not significant when brittle materials
are impacted by rounded grains [Head and Harr, 1970], as
is generally the case in bedrock rivers. Note also that this
formulation implicitly assumes that the mass of impacting
particles is conserved, because it does not account for the
energy that may contribute to wear of the impacting grain or
for the kinetic energy of any fragments that may be
produced by the grain impact.
[18] The linear dependence of impact abrasion rate on
excess energy is illustrated in Figure 4 by the experimental
Figure 3. Saltation trajectory definition sketch. Shown are
the saltation hop height (Hs), mean downstream sediment
velocity (us), mean vertical descent velocity (wsd), horizontal (usi) and vertical (wsi) impact velocities, and the
upward (Lsu) and downward (Lsd) portions of the total hop
length (Ls).
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comparison of model predictions with the experimental
data (section 5). Thus, for erosion by bed load sediment
moving by saltation, equation (3) can be rewritten as
Vi ¼
prs D3s w2si Y
6kv s2T
ð5Þ
where prsD3s /6 = Mp, rs is the density of sediment, Ds is the
diameter of a spherical sediment grain, and wsi is the
vertical component of the particle velocity on impact (wsi =
Ui sin a) (Figure 3).
Figure 4. Variation in volume eroded with net energy
transferred per unit mass of impacting particles. Data are
from Head and Harr [1970] for glass plate eroded by glass
beads for normal (a = 90) and oblique (a = 15) impact
angles.
data of Head and Harr [1970], who eroded brittle plate
glass with a stream of air-blown 0.04-mm diameter glass
beads. In these experiments, they varied the energy flux by
changing the air velocity and determined the net energy
transferred by measuring particle impact and rebound
velocities with high-speed photography. Similarly, the
influence of impact angle on abrasion rates is shown in
Figure 5. Here data from Head and Harr [1970] are plotted
along with results from experiments by Bitter [1963a] in
which round 0.3-mm diameter iron pellets were dropped on
plate glass from a height of five meters. Note that Head and
Harr’s [1970] results are consistent with a negligible
erosion threshold, unlike Bitter’s [1963a] data, which do
indicate an erosion threshold possibly due to the use of
ductile rather than brittle abrasive material.
[19] The resistance of rock and other brittle materials
to abrasion by impacting particles (parameterized in
equation (3) by ev) depends on the capacity of the material
to store energy elastically [Engle, 1978]. Referred to
variously as the material toughness, resilience or strain
energy, the capacity to store energy elastically (b) is defined
quantitatively as the area under the stress-strain curve at the
yield stress, so that
ev ¼ kv b ¼ kv
s2T
2Y
2.4. Particle Impact Rate (Ir)
[20] The rate of particle impacts, per unit time and per
unit area (Ir), should be linearly proportional to the flux of
bed load particles and inversely proportional to the downstream distance between impacts. The particle flux is simply
the mass flux per unit width (qs) divided by the mass per
particle, and the distance traveled between impacts is the
saltation hop length (Ls), so that particle impact rate can be
written as
Ir ¼
6qs
:
prs D3s Ls
ð6Þ
2.5. Fraction of Bed Exposed (Fe)
[21] When sediment mantles portions of the bedrock bed,
some of the bed load kinetic energy acquired from the
surrounding flow is expended in collisions with immobile
sediment rather than exposed bedrock. Because transient
sediment deposits are common in actively incising rivers,
we assume a fraction of saltation hops occur as bed load
across local patches of alluvium. The kinetic impact energy
is thus lost to intergranular friction, without contributing to
bedrock fracture growth, except perhaps when the alluvial
cover is only one or two grain diameters thick. Additional
energy is lost in collisions between mobile grains.
[22] We parameterize the effect of local alluvial cover and
grain-to-grain interaction as a function of the sediment
transport capacity in excess of the sediment supply. Consider the end-member cases. When the channel has just
enough transport capacity to move the coarse sediment
supplied from upstream and from local hillslopes, the bed
should tend to be fully alluviated. Any increase in the rate of
bed load sediment supply (assuming no change in grain
ð4Þ
where sT is the rock tensile yield strength, Y is Young’s
modulus of elasticity and kv is a dimensionless coefficient
that will depend in part on the material properties of the
impacting particle. Because the variation in modulus of
elasticity of rocks is limited [Clark, 1966], to first order Y
can be treated as constant. Thus bedrock abrasion resistance
scales directly with the square of rock tensile strength, as
demonstrated in the bedrock abrasion mill experiments of
Sklar and Dietrich [2001]. In principle, the erosion
threshold parameter et should also scale with rock tensile
strength, however, the abrasion mill data [Sklar and
Dietrich, 2001] suggest that the energy threshold et can
be neglected in the case of bed load abrasion because even
the relatively low-impact energy of fine sand moving as bed
load is sufficient to cause measurable bedrock wear. We
henceforth set et = 0, an assumption supported by
Figure 5. Variation in erosion rate with normal component
of energy flux, represented by sin2(a). Erosion rates are
normalized by maximum for each experiment.
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Table 1. Experimental Conditions for Saltation Trajectory Studies
Abbott and Francis [1977]
Fernandez Luque and van Beek [1976]
Francis [1973]
Hu and Hui [1996a, 1996b]
Lee and Hsu [1994]
Nino et al. [1994]
Sekine and Kikkawa [1992]
Wiberg and Smith [1985]
Ds, mm
rs, Mg/m3
Rep
Fr
Material
Bed
Shape
8.8
3.3
4.6 – 9.8
2.3 – 3.5
1.4, 2.5
15, 31
5, 10
5.0 – 8.0
2.6 – 2.9
2.64
2.65
2.2 – 2.6
2.5
2.65
2.5
2.5 – 2.6
500 – 900
120 – 160
300 – 1000
180 – 420
100 – 220
2000 – 5000
370 – 1800
300 – 1300
na
>1.0
0.8 – 1.5
0.5 – 1.5
1.6 – 2.0
1.0 – 1.5
na
na
natural
natural
natural
glass, quartz
natural
natural
glass
na
fixed
fixed
fixed
fixed and moveable
fixed
moveable
moveable
splash function
rounded pea gravel
na
water-worn gravel
spherical and irregular
na
na
beads
ellipsoidal
size) will cause aggradation until a steeper slope is achieved
so that transport capacity again equals the sediment supply
rate. In contrast, when the channel receives no coarse
sediment supply, as might occur in mechanically weak
lithologies that weather rapidly to suspended load size
particles, no coarse sediment is available to form patches
of alluvial cover, and bedrock should be exposed across the
entire bed. Here we assume that the fraction of the bed
composed of exposed bedrock (Fe) varies linearly between
these end-members, so that
Fe ¼ ð1 qs =qt Þ
ffor qs qt g
ð7Þ
where qt is the sediment transport capacity for a fully
alluviated bed. In subsequent calculations we use the
Fernandez-Luque and van Beek [1976] bed load sediment
transport relation
1=2
ðt* tc*Þ3=2
qt ¼ 5:7rs Rb gD3s
ð8Þ
where qt is the sediment mass transport capacity per unit
width, rs is the sediment density, Rb is the nondimensional
buoyant density of sediment (Rb = rs/rw 1), rw is the
density of water, g is the acceleration due to gravity, t* =
tb/[(rs rw)gDs] is the nondimensional form of the
boundary shear stress (tb), and t*c is the value of t* at the
threshold of particle motion.
[23] The linear assumption of equation (7) is supported
by flume experiments [Sklar and Dietrich, 2002] in which
we observe a generally linear increase in bed cover with
increasing sediment supply. The experiments also demonstrate the potential for more complex behavior, such as a
threshold supply rate below which alluviation does not
occur and instabilities associated with changing bed roughness, particularly for the case of smooth, planar bedrock
channels. We do not attempt to model such nonlinear
behavior, in effect assuming that it will be suppressed by the
irregular topography typical of bedrock channel beds.
2.6. Composite Expression
[24] Our simple expression for bedrock abrasion rate, E =
ViIrFe, can now be restated by substituting equations (5),
(6), and (7) into equation (1) and simplifying, resulting in
the composite expression for bedrock incision rate
qs w2si Y
E¼
Ls kv s2T
qs
:
1
qt
ð9Þ
3. Saltation Trajectories
[25] In this section we develop expressions for the saltation hop length (Ls) and the vertical component of the
particle impact velocity (wsi), so that we can express
bedrock abrasion rate (equation (9)) directly in terms of
shear stress. The most mechanistically rigorous approach
would be to build a model for saltation over both bedrock
and alluvial beds that solves the equations of motion for an
individual grain throughout the saltation trajectory. Theoretical models have been developed for saltation in water over
an alluvial bed [Wiberg and Smith, 1985; Sekine and
Kikkawa, 1992] and for saltation in air [e.g., Anderson and
Haff, 1988]. Such models need to account for the forces due
to hydrodynamic lift and drag, gravity, the virtual mass
effect (caused by the relative accelerations of the fluid and
particle), the spinning motion of saltating grains, and
collisions with the bed [Wiberg and Smith, 1985]. This level
of complexity is unnecessary for our purposes here. Instead,
we use a large set of published data to obtain a set of
empirical expressions that together constitute a selfconsistent description of how saltation trajectories vary as
a function of shear stress and grain size.
3.1. Data Sources
[26] The pioneering experimental saltation studies of
Francis [1973], Fernandez Luque and van Beek [1976],
and Abbott and Francis [1977] produced an important set of
observations and data that has helped guide the development of bed load transport theory [e.g., Bridge and
Dominic, 1984]. More recently, the available experimental
data set has grown considerably with studies by Sekine and
Kikkawa [1992], Lee and Hsu [1994], Nino et al. [1994],
and Hu and Hui [1996a, 1996b]. In each of these studies,
measurements of saltation trajectories were obtained from
high-speed photographs or video images. In addition to
these seven experimental data sets, we also make use of
saltation trajectories predicted by the theoretical model of
Wiberg and Smith [1985]. A wide range of experimental
conditions are represented in these studies, including fixed
and moveable beds, natural and artificial particles, and
supercritical and subcritical flow. We limit our analysis to
measurements made under hydraulically rough conditions
(i.e., Rep > 100, where Rep = rwDsu*/m is the particle
Reynolds number, u* = (tb/rw)1/2 is the shear velocity, and m
is the kinematic viscosity). This has the effect of excluding
trajectories of grains with either small diameters (Ds <
1.4 mm) or low buoyant density (Rb < 1.2). Table 1 lists the
experimental conditions for each data set used.
[27] Each study reports measurements of the three basic
saltation trajectory components, mean downstream velocity
(us), hop height (Hs), and hop length (Ls), except for Francis
[1973] and Fernandez Luque and van Beek [1976], who
only report downstream velocity. The variation of each
component is described as a function of flow intensity,
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Table 2. Saltation Trajectory Data Conversiona
us(y)
Abbott and
Francis [1977]
Fernandez Luque
and van Beek
[1976]
Francis [1973]
Hu and Hui
[1996a, 1996b]b
Lee and Hsu [1994]
Nino et al. [1994]
Se kine and
Kikkawa [1992]c,d
Wiberg and
Smith [1985]
us(x)
us Source
Hs(y)
Hs(x)
Hs Source
Ls(y)
Ls(x)
Ls Source
t*c
t*c Source
us
u*
Figure 11
Hs
u*/u*c
Figure 5
Ls
u*/u*c
Figure 5
0.03
assumed
us
u* 0.7 u*c
Figure 6
na
na
na
na
na
na
0.047
Figure 2
us
us/u*
u*
(t*c /t*)0.5
Appendix 2
Figure 12
na
Hs/Ds
na
t*
na
Figure 10
na
Ls/Ds
na
t*
na
Figure 11
0.05
0.03
Table 7
Figure 2
us/u*
t* t*c
Table 2
Hs/Ds
t* t*c
Table 2
Ls/Ds
t* t*c
Table 2
Table 2
us/u*
us/(RbgDs)0.5
t*/t*c
u*/wf
Figure 4
Figure 10
Hs/Ds
Hs/Ds
t*/t*c
t*
Figure 3
Figure 9
Ls/Ds
na
t*/t*c
na
Figure 3
na
0.026,
0.031
0.036
0.03
Figure 8
assumed
us, us/u*c
t*/t*c
Tables 1
and 2
Hs/Ds
t*/t*c
Tables 1
and 2
Ls/Ds
t*/t*c
Table 1, 2
0.06
Figure 3
a
X(y) and X(x) refer to y axis and x axis variables in original reporting of data.
Hu and Hui [1996a, 1996b] report a discrepancy between values of u*, Ds, rs and t* in their Table 1 corrected by dividing t* by specific density, Hs and
Ls data sets only.
c
Fall velocity calculated for Sekine and Kikkawa [1992] data using method of Dietrich [1982] with CSF = 0.8 and Powers scale of 3.5.
d
Sekine and Kikkawa [1992] report anomalously long hop lengths, possibly due to gravity component on steep slope (G. Parker, personal
communication). Data are not included.
b
represented either by nondimensional shear stress (t*) or
shear velocity (u*). No study reports direct measurements
of the vertical component of the impact velocity (wsi),
one of the two saltation trajectory components we seek to
predict. We can, however, estimate wsi from the mean
particle descent velocity (wsd), which depends on the
three basic saltation trajectory components, us, Hs and Ls
(Figure 3).
[28] Rather than select the results of any single study as
the best general representation of saltation trajectories, we
choose to find relationships that best fit the trends of the
data taken as an ensemble. This requires finding the
nondimensionalization that provides the best collapse of
the data for each variable. These algebraic manipulations
require no additional information except in two cases, the
data of Abbott and Francis [1977] and Sekine and
Kikkawa [1992] for which we must assume a value for
t*c. We chose t*c = 0.03 because a low value is appropriate
for motion over fixed beds and for grains introduced into
the flow from above [Buffington and Montgomery, 1997].
Table 2 provides details on the data source and conversion
information for each data set.
stress (t*/t*c 1) so that velocity goes to zero at the
threshold of motion. Regressing the data in this manner
(Figure 7a), we obtain
us
ð Rb gDs Þ0:5
0:56
t*
1
¼ 1:56
:
tc*
ð11Þ
Equation (11) is plotted in Figure 6 as the thick line.
3.2. Mean Downstream Saltation Velocity (us)
[29] We find the best data collapse when we nondimensionalize us by dividing by (RbgDs)1/2. Figure 6 shows the
variation in mean downstream saltation velocity with
transport stage (t*/t*c). The best fit log-log linear regression
line through the data ensemble is
us
ðRb gDs Þ0:5
0:83
t*
¼ 0:83
:
tc*
ð10Þ
Goodness of fit for this and subsequent regressions is
indicated by the standard error of the fit parameters and is
shown in Figures 6, 7, and 9. Note that the regression fit
predicts a nonzero mean downstream velocity at the
threshold of motion. A more physically consistent approach
is to treat us as a function of nondimensional excess shear
Figure 6. Variation in downstream saltation velocity with
transport stage. Data are taken from seven published
experimental studies and one theoretical model [Wiberg
and Smith, 1985]. Thin and thick lines represent results of
log-log linear regressions of saltation velocity with transport
stage and excess shear stress respectively.
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reduced intensity of particle motion at low excess shear
stresses, and to be consistent with bed load sediment
transport capacity relations (e.g., equation (8)), which
predict zero sediment transport at the threshold-of-motion
shear stress.
3.3. Saltation Hop Height (Hs)
[31] Figure 7b shows the variation in hop height with
excess shear stress, nondimensionalized by particle diameter. Maximum observed hop heights are about five particle
diameters, for large excess shear stresses. The log-log linear
regression line that best fits the data ensemble is
0:50
Hs
t*
1
¼ 1:44
:
tc*
Ds
ð12Þ
3.4. Saltation Hop Length (Ls)
and the Transition to Suspended Transport
[32] Figure 7c shows the variation in saltation hop length
with excess shear stress, nondimensionalized by particle
diameter. The log-log linear best fit relationship is
0:88
Ls
t*
1
¼ 8:0
:
tc*
Ds
Figure 7. Variation in principal saltation trajectory
components with excess shear stress. Symbols are the
same as in Figure 6. (a) Downstream saltation velocity.
(b) Saltation hop height. (c) Saltation hop length.
Uncertainty in log-log linear regression equations is
standard error of the parameter estimates. Thick line in
Figure 7c is nonlinear extrapolation assuming hop lengths
become infinite as transport mode shifts from bed load to
suspended load.
[30] Equation (11) better represents the trend in the data
because there is no strong curvature in the residuals, unlike
equation (10); note also that the uncertainty in the parameter estimates is lower. Equation (11) is also more consistent with the fact that rolling and sliding occur in the
transition between no motion and full saltation. The vast
majority of grains move in saltation mode once t*/t*c
reaches about 1.2 [Abbott and Francis, 1977; Hu and
Hui, 1996a]. Therefore we henceforth treat downstream
velocity and the other components of the saltation trajectory
as functions of t*/t*c 1 (Figure 7) to account for the
ð13Þ
This relationship, however, should not hold for very large
excess shear stresses because of the transition from bed load
to suspended particle motion. Equations (11) and (12)
similarly should not apply when the transport mode shifts
from bed load to suspended load. The detailed observations
of Abbott and Francis [1977] show that rising turbulent
eddies interrupt the downward acceleration of descending
saltating grains even at low excess shear stresses. As shear
stress increases, such ‘‘suspensive’’ saltation hops become
more frequent. Full suspended transport, when the upward
diffusion of momentum balances gravity, occurs when the
shear velocity (u* = (tb/rw)1/2) exceeds the particle fall
velocity in still water (wf) [Rouse, 1937]. The transition
from bed load to suspension is equivalent to a grain taking a
hop of infinite length.
[33] To account for the shift in the mode of particle
transport, we divide equation (13) by a nonlinear function
of u*/wf so that hop length grows rapidly as u* approaches
wf and becomes infinite at the threshold of suspension
Ls
8:0ðt*=tc* 1Þ0:88
¼
2 1=2
Ds
1 u*=wf
for
u*=wf < 1:
ð14Þ
This nonlinear extrapolation is shown as the thick line in
Figure 7c. Note that significant deviation from the log-log
linear regression fit occurs only above excess shear stresses
of about 20. The fact that no experimental data exist for any
of the three saltation trajectory components at excess shear
stresses above this range is consistent with a shift from
saltation to suspended motion.
[34] The transport stage that corresponds to the threshold
of suspension (i.e., the value of t*/t*c when u*/wf =
(tb/rw)1/2/wf = 1) and thus the excess shear stress at which
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elongated with increasing excess shear stress; thus saltating
grains impact less frequently per unit distance traveled as
excess shear stress increases.
3.6. Vertical Impact Velocity (wsi)
[36] No direct measurements of the vertical component of
particle impact velocity (wsi) are available in the published
experimental studies. However, the variation in wsi with
excess shear stress should scale with the mean particle
descent velocity (wsd). We can calculate wsd as simply the
hop height (Hs) divided by the descent time (td)
wsd ¼
Figure 8. Particle fall velocity in still water (wf) and
transport stage at the threshold of suspension (t*/t*c )susp as
functions of particle diameter. Fall velocity was calculated
using the method of Dietrich [1982].
w2f
t*
¼
t*c susp t*c Rb gDs
ð15Þ
because of the dependence of fall velocity on grain size.
Dietrich [1982] provided an empirical method for calculating fall velocity as a function of grain size for varying grain
shape (Cory shape factor) and angularity (Powers scale). We
calculate wf using this method, assuming values of Cory
shape factor (0.8) and Powers scale (3.5) typical for natural
gravel grains [Dietrich, 1982]. Figure 8 shows fall velocity
and transport stage at the threshold of suspension plotted
against grain size. For grain sizes less than about 0.0001 m,
the threshold of suspension occurs at the threshold of
motion (t*/t*c = 1), thus silt and clay sized grains travel as
wash load. The peak in the (t*/t*c)susp curve is due to the
change in flow around the falling grain, from hydraulically
smooth flow for fine grains to hydraulically rough flow for
coarse grains.
3.5. Nondimensional Saltation Trajectory
[35] The ratio of hop length to hop height is a nondimensional measure of the shape of the saltation trajectory. We
can use this ratio to test whether equations (12) and (14) fit
not only the individual components of the trajectory but the
overall trajectory shape as well. We do this by comparing
the trajectory shape predicted by the ratio of equations (12)
and (14)
Ls
5:6ðt*=t*c 1Þ0:38
¼
2 1=2
Hs
1 u*=wf
ð17Þ
where (td ffi Lsd/us), and Lsd and Lsu are the descending and
rising portions of the hop length respectively (Figure 3). Hu
and Hui [1996b] report measurements of Lsu and Ls that
indicate a linear relationship [Sklar, 2003], which we
approximate here as
Lsd ¼ 2=3 Ls :
saltation hop length becomes infinite, depends nonlinearly
on grain size
Hs Hs us
Hs us
¼
td
Lsd
Ls Lsu
ð18Þ
Combining equations (11), (12), (14), (17), and (18) we
obtain an expression for mean particle descent velocity
wsd ¼
2 1=2
3Hs us
¼ 0:4ðRb gDs Þ1=2 ðt*=t*c 1Þ0:18 1 u*=wf
2Ls
ð19Þ
Saltating grains accelerate as they fall and probably reach
their maximum descent velocity at the end of the hop. From
the scaled plots of individual saltation trajectories of Abbott
and Francis [1977, Figure 13] and from the modeled
vertical velocity reported by Wiberg and Smith [1985,
Figure 15], we estimate that vertical impact velocity can be
approximated as twice the mean descent velocity
wsi 2wsd :
ð20Þ
3.7. Calculating Bedrock Incision Rates
[37] Using expressions for saltation hop length (equation
(14)), vertical impact velocity (equation (20)), and bed load
ð16Þ
to the subset of data where measurements of hop length and
height were taken from the same hop event. This is shown
in Figure 9, along with the best fit log-log linear line
through the data pairs. We conclude that equations (12)
and (14) represent the saltation trajectory shape well.
Importantly, note that saltation trajectories grow more
Figure 9. Variation in ratio of saltation hop length to hop
height with excess shear stress. Symbols are the same as in
Figure 6.
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sediment transport capacity (equation (8)), we can now
express the three component terms of the saltation-abrasion
bedrock incision model directly in terms of excess shear
stress
0:36
2 !
t*
u*
1
1
tc*
wf
0:88
3qs
t*
us
1
1
4prs D4s tc*
wf
Fe ¼ 1 ð21Þ
2 !0:5
ð22Þ
qs
0:5
5:7rs Rb gD3s tc* 3
1:5
t*
1
:
tc*
ð23Þ
Multiplying these three terms, rounding the resulting
exponent on excess shear stress to one decimal place (i.e.,
0.52 0.5), and simplifying we obtain a final form of
the saltation-abrasion bedrock incision model
2 !3=2
1=2 0:08Rb gY
t*
qs
us
1
E¼
qs
1
1
2
qt
wf
kv st
tc*
ð24aÞ
for qs qt and u* wf (E = 0 for qs > qt or for u* > wf). If
we neglect the suspension effect term, or equivalently
consider only excess shear stresses well below the threshold
of suspension{i.e., 0 < (t*/t*c 1) < 10}, and substitute
equation (8) for qt and rearrange, we obtain a simpler
expression for bedrock abrasion rate
E ¼ k1
qs
1=2
ðt*=tc* 1Þ
k2
q2s
3=2
Ds ðt*=tc*
1Þ2
ð24bÞ
where the tools effect and cover effect are represented by
the first and second terms respectively, k1 = 0.08YRbg/kvsT,
k2 = 0.014Y(Rbg)0.5/(t*c)1.5kvs2t rs, valid for qs qt.
[ 38 ] In the model results reported below, we use
equation (24a) to calculate bedrock incision rates from
seven input variables: channel slope (S), channel width
(W), total sediment supply rate (Qs = qsW), discharge (Qw),
grain size (Ds), a channel roughness parameter (n), and
rock tensile strength (sT). Average boundary shear stress
(tb) is calculated using the hydraulic radius (rh) in place
of flow depth (Hw) to account for drag due to the channel
banks
tb ¼ rw grh S
ð25Þ
where rh is found using the Manning equation for mean
flow velocity uw = rh2/3S1/2/n and assuming a rectangular
channel cross section
rh ¼
kv, the dimensionless proportionality constant relating rock
resistance (ev) to elastic strain energy (b = s2T/2Y) in
equation (4). In section 6 we use experimental measurements of bedrock wear [Sklar and Dietrich, 2001] to
obtain an order of magnitude estimate for kv of 1012.
pr Rb gD4s Y
Vi ¼ s
9:4kv s2T
Ir ¼
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Hw W
¼
2Hw þ W
Qw n
Hw WS 1=2
3=2
:
ð26Þ
All other quantities (rs, rw, g, t*c, and Y) are considered
constants. Note that the only remaining free parameter is
4. Model Results
[39] The saltation-abrasion model (equation (24)) suggests that rates of bedrock incision by saltating bed load
should depend, for a given rock strength, on three principal
variables: shear stress, sediment supply, and grain size. In
this section, we explore the behavior of the saltationabrasion bedrock incision model as we vary each of those
three principal variables independently. We then use a
nondimensional form of the model to show that the full
model behavior can be depicted compactly in a single
nondimensional graph. Note that in all the calculations
reported below we have been careful not to violate the
assumptions underlying the development of the model, for
example flow depths are always greater than the grain
diameter and particle Reynolds numbers remain in the
hydraulically rough regime.
[40] To help guide our selection of model input values
we use a particular reach of bedrock river as a reference
site, a gauged reach of the South Fork Eel River in
northern California (Figure 1) [Seidl and Dietrich, 1992;
Howard, 1998]. This actively incising, mixed alluvialbedrock bedded canyon river motivated many of the
assumptions underlying development of this model. We
use the reference site as a device to ensure a physically
reasonable set of values for the variables held constant as
we conduct the following model sensitivity analysis, and
to provide perspective on the extent of the range of
possible input values encompassed by the model parameter space.
[41] Table 3 lists the reference site values for channel
geometry, roughness, rock strength, discharge, sediment
supply, and grain size used as model inputs in the
following sensitivity analysis. Also listed are some of
the calculated hydraulic conditions, such as flow depth
and mean velocity, and model output for the reference
site. The choice of values of discharge and sediment
supply to represent the reference site is developed fully
by Sklar [2003]. The reference discharge of 39.1 m3/s has
an exceedence probability of 0.013 for mean daily flow,
based on analysis of a 24 year record of daily mean
discharges, USGS gauge 11475500 ‘‘S.Fork Eel near
Branscom, CA.’’ A 24 hour duration of the reference bed
load sediment supply of 42.6 kg/s would represent 6% of
the annual coarse sediment yield to the reach assuming an
average landscape lowering rate of 0.9 mm/yr [Merritts and
Bull, 1989] and that 22% of the total load is in the bed load
size range.
[42] We assume here that the net effect of the full
distribution of discharge and sediment supply events in
transporting sediment and eroding bedrock, can be
accomplished by the reference discharge and sediment
supply acting over a limited duration, which in this
case is approximately 22 days per year. We assume that
flows too low to mobilize sediment (and thus erode
bedrock) occur on the remaining 343 days. For reference site conditions, the saltation-abrasion model pre-
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Table 3. Reference Site Model Input and Output Valuesa
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13, 14, and 15), the curve passing through the reference site
values is plotted with a thick solid line; an open circle
indicates the reference site.
Value
Model Inputs
Channel slope S
Discharge Qw
Grain diameter Ds
Sediment supply Qs
Channel width W
Roughness n
Rock tensile strength sT
Rock elastic modulus Y
Dimensionless rock resistance parameter kv
Nondimensional critical shear stress t*c
Sediment density rs
Water density rw
0.0053b
39.1 m3/sc
0.060 mb
42.6 kg/sc
18.0 mb
0.035d
7.0 MPae
5.0 104 MPad
1.0 1012 e
0.030d
2650 kg/m3 d
1000 kg/m3 d
Model Outputs
Instantaneous erosion rate E
Flow depth Hw
Mean flow velocity uw
Froude number Fr
Transport stage t*/t*c
Sediment transport capacity Qt
Ratio of shear velocity to fall velocity u*/wf
Boundary shear stress tb
20.5 mm/yr
1.1 m
2.0 m/s
0.62
1.7
50.2 kg/s
0.24
53.7 Pa
dicts an instantaneous bedrock incision rate of 20 mm/yr
(0.056 mm/day), which is equivalent to an annual average
incision rate of 0.9 mm/yr. Note that in each of the figures
showing model results in this section (Figures 10, 11, 12,
4.1. Influence of Sediment Supply
[43] Figure 10 shows how incision rate predicted by
the saltation-abrasion model varies with sediment supply,
holding all other input variables to constant reference
site values. Also plotted are the component terms from
equation (1), the volume eroded per impact (Vi), the rate of
impacts by saltating grains (Ir), and the fraction of bed area
exposed to abrasive wear (Fe). We find a parabolic
variation in erosion rate, as expected from equation (24).
At low supply rates, incision rate grows with increasing
sediment supply, reflecting the growth in impact rate. At
high supply rates, incision rate decreases with increasing
supply rate, due to the reduction in the fraction of bed
area exposed. Erosion rate is zero when the sediment
supply equals the transport capacity and the bed is fully
alluviated, and in the trivial case when there is no
sediment supply. The maximum incision rate corresponds
to a critical level of sediment supply where the growth in
impact rate is balanced by the reduction in fraction of bed
area exposed. Because shear stress and grain size are not
varied here, the volume eroded per impact remains
constant.
[44] If we repeat this calculation for different values of
channel slope, and thus different shear stresses, we generate
a set of parabolic erosion rate curves, as shown in Figure 11.
As shear stress increases from the threshold of motion, the
peak erosion rate increases along with the sediment supply
rate required to alluviate the bed and shut off incision. As
transport stage rises above a value of about 10, however, the
magnitude of the peak erosion rate begins to decline. This
occurs because the increase in particle impact energy with
greater excess shear stress is more than offset by the
Figure 10. Erosion rate as a function of sediment supply
predicted by saltation-abrasion model (equation (24a)). Also
plotted are the model component terms: volume eroded per
unit impact (Vi), impact rate per unit area (Ir), and the
fraction of the bedrock bed exposed (Fe). Table 3 lists the
values of input variables held constant. The open circle
denotes the predicted instantaneous erosion rate at the South
Fork Eel reference site.
Figure 11. Erosion rate as a function of sediment supply
for various channel slopes, with corresponding transport
stage noted. Thick solid line corresponds to ‘‘reference site’’
conditions (Table 3) and curve of Figure 10.
a
These values are used in model calculations unless otherwise specified.
Values for channel conditions and water and sediment fluxes are intended to
approximate a gauged reach of the South Fork Eel River, Mendocino
County, California. Also listed are values of model output for reference site
conditions.
b
Field survey.
c
Magnitude frequency analysis [Sklar, 2003].
d
Assumed.
e
Laboratory measurement.
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Figure 12. Erosion rate as a function of transport stage
predicted by saltation-abrasion model. Also plotted are the
model component terms: volume eroded per unit impact
(Vi), impact rate per unit area (Ir) and the fraction of the
bedrock bed exposed (Fe).
reduction in frequency of particle impact as the saltation
hop length rapidly lengthens.
4.2. Influence of Transport Stage
[45] The dependence of erosion rate on excess shear
stress is more clearly illustrated in Figure 12, where we
again plot erosion rate along with the component terms from
equation (1), now holding sediment supply and other
variables constant. Erosion rate is zero below the transport
stage required to transport the supplied sediment load. Even
though transport stage exceeds the threshold of motion, the
bed is assumed fully alluviated. Erosion rate increases
rapidly once there is excess sediment transport capacity,
reflecting the growth in both the extent of bedrock exposure
in the channel bed and the particle impact energy. The
erosion rate peaks due to a decline in the rate of growth in Vi
and Fe, so that these two terms no longer offset the
continuous reduction in impact frequency (plotted separately on a logarithmic scale) with increasing shear stress.
Erosion rates go to zero for high-transport stage as the
saltation hop length approaches infinity at the threshold of
suspension. Note that the peak erosion rate occurs at an
excess shear stress well within the range of experimental
saltation data used to parameterize the model, and is not an
artifact of the extrapolation we use to account for the
transition from bed load to suspended transport mode
(equation (14)). The peak in erosion rate with increasing
shear stress (hereafter called the ‘‘transport peak’’ as
distinguished from the ‘‘supply peak’’ discussed above)
occurs at a relatively low transport stage in Figure 12
because sediment supply is held constant. The transport
peak would occur at a higher excess shear stress if we
allowed sediment supply to increase along with shear stress.
[46] Figure 13 illustrates the variation in incision rate
with shear stress for various levels of sediment supply,
holding the remaining variables constant. Greater rates of
sediment supply require greater excess transport capacity to
expose bedrock and initiate incision (i.e., for qt > qs). Peak
erosion rates occur at greater transport stages for larger
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Figure 13. Erosion rate as a function of transport stage for
various rates of sediment supply. Thick solid line
corresponds to ‘‘reference site’’ conditions and curve of
Figure 12.
sediment supply rates. The magnitude of the peak erosion
rate rises and then declines with increasing sediment supply
rate. This is similar to the pattern of the erosion rate peaks
shown in Figure 11, except that now the roles of sediment
supply and shear stress are reversed. Erosion rate goes to zero
in all cases as transport stage approaches the threshold of
suspension, which for Ds = 0.06 m occurs when t*/t*
c ffi 30.
4.3. Influence of Grain Size
[47] Another way to understand the variation in erosion
rate with shear stress is to vary transport stage by varying
grain size, holding slope and all other variables constant.
This is shown for reference site conditions in Figure 14,
where we again plot Vi, Ir and Fe, the component terms of
equation (1), as well as the transport stage. Here high-
Figure 14. Erosion rate as a function of grain diameter.
Also plotted are the model component terms: volume
eroded per unit impact (Vi), impact rate per unit area (Ir),
and the fraction of the bedrock bed exposed (Fe).
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transport stage corresponds to small grain size, because Ds
is in the denominator of t*. Erosion rate is zero for grain
sizes greater than 0.065 m because transport capacity drops
below sediment supply rate. Erosion rate is zero for grain
sizes smaller than about 0.003 m because volume eroded
per impact (Vi) and particle impact rate (Ir) are zero when
grains are in suspension. For grain sizes above 0.003 m, Ir is
initially high and Vi initially low because the very small
mass of an individual particle results in a large number of
particles, for a fixed sediment supply rate, each with very
low kinetic energy. As grain size increases, Vi grows more
rapidly than Ir declines, such that the product of Vi and Ir,
equivalent to the net flux of kinetic energy, grows larger.
Erosion rate peaks at the grain size where the rate of
reduction in bed exposure (Fe) equals the rate of increase in
kinetic energy flux, and declines to zero as the reduction in
excess shear stress approaches the threshold of alluviation.
Once Fe = 0 (when Qs = Qt), erosion rate is zero, even
though larger grains would still be in motion and thus have
a nonzero impact energy. Note that as the threshold of
motion is approached the decline in particle impact rate (Ir)
with increasing grain size is reversed and at the threshold of
motion the impact rate is infinite. This singularity in
equation (23) is physically equivalent to a transition to
rolling motion, but does not affect predicted erosion rates
because the threshold of alluviation always occurs at a
higher excess shear stress than the threshold of motion.
Figure 14 is conceptually a mirror image of Figure 12
because the direction of increase in transport stage along the
horizontal axis is reversed. Note that the curve shapes are
not exact mirror images because linear variation in grain
size is equivalent to hyperbolic variation in transport stage.
[48] The role of grain size in controlling sediment transport capacity for a given slope and discharge and thus
influencing the extent of bedrock exposure is illustrated in
Figure 15, where we plot erosion rate as a function of grain
size for various sediment supply rates. Irrespective of
sediment supply, erosion rate is zero for grains too small
to move as bed load. As in Figure 14, erosion rate rises with
increasing grain size to a peak and then declines to zero as
the sediment transport capacity approaches the sediment
supply rate. A maximum in the peak erosion rate occurs for
an intermediate level of sediment supply, where the tools
and coverage effects are balanced. Note that for larger
sediment supply rates, full bed alluviation occurs at smaller
grain sizes, in effect narrowing the range of grain sizes
capable of eroding bedrock. Figure 15 is conceptually a
mirror image of Figure 13; each illustrates the effect of
variable transport stage on bedrock erosion rates.
4.4. Nondimensional Framework
[49] The predicted dependence of bedrock abrasion rate
on sediment supply and shear stress can be described
compactly by expressing the saltation abrasion model in a
nondimensional form. We define a dimensionless sediment
supply (q*s ) as the sediment supply (qs) relative to the
transport capacity (qt)
qs* ¼
qs
:
qt
ð27Þ
Substituting equation (27) and the bed load sediment
transport relation (equation (8)) into equation (24a) and
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Figure 15. Erosion rate as a function of grain diameter for
various rates of sediment supply. Thick solid line
corresponds to ‘‘reference site’’ conditions and curve of
Figure 14.
rearranging, we obtain an expression for a nondimensional
bedrock incision rate (E*)
2 !3=2
t*
u*
1 1
E* ¼ k3 qs*ð1 qs*Þ
t*c
wf
ð28Þ
where E* = Es2T/rsY(gDs)3/2 and k3 = 0.46R3/2
b t*
c . Because
the ratio u*/wf is a function of transport stage for a given
grain size (Figure 8), the nondimensional erosion rate (E*)
can be considered a function of just two nondimensional
quantities, the relative sediment supply (q*s) and the transport
stage (t*/t*c). An equivalent nondimensional functional
relationship can be derived by formal dimensional analysis
using the Buckingham pi theorem [Sklar, 2003].
[50] Using this nondimensional framework the saltationabrasion model is plotted in Figure 16, which shows
contour lines of equal dimensionless erosion rate (E*,
contour labels 1015) as a function of relative sediment
supply and transport stage. As discussed above, relative
sediment supply is assumed to be equal to the fraction
of bed area covered by transient alluvial deposits (i.e., q*s =
1 Fe). Using this framework, the saltation-abrasion model
collapses to a unique functional surface for all physically
reasonable combinations of the seven model inputs: sediment
supply (Qs), coarse sediment grain size (Ds), discharge (Qw),
slope (S), channel width (W), channel roughness (n), and rock
tensile strength (sT). The surface has single peak value of
E* = 7.7 1015, which occurs at q*s = 0.50 and t*/t*c ffi 12.
[51] Erosion rate goes to zero at each of the four boundary conditions. Along the transport stage axis, erosion rate is
zero at the threshold of motion and at the threshold of
suspension. Along the relative supply axis erosion rate is
zero when the bed is fully alluviated and when the bed is
composed of bare bedrock due to a negligible supply of
tools. The peak nondimensional erosion rate occurs when
the tradeoffs between the cover and tools effects, and
between the energy and frequency of particle impacts, are
exactly balanced.
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excess shear stress, which for compactness we represent
here by g (i.e., g = t*/t*c 1),
dE
2k2 q2
k1 qs
¼ 3=2 s 3=2 :
dg Ds g3 2g
ð29Þ
Equation (29) equals zero when
qs ¼
Figure 16. Nondimensional erosion rate (E*) 1015
(contours, axis out of view) as a function of transport
stage and relative sediment supply. In this nondimensional framework the saltation-abrasion model collapses
to a unique surface for all physically reasonable
combinations of discharge, channel slope, width, roughness, rock tensile strength, and coarse sediment grain
size (Ds > 0.002 m). Erosion rate goes to zero as each
of the four end-member conditions are approached: the
thresholds of motion and suspension and a fully alluvial
or pure bedrock bed.
[52] Figure 17 shows how the three curves corresponding
to reference site conditions in Figures 11, 13, and 15 are, in
effect, vertical slices through the nondimensional erosion
function surface. Varying sediment supply while holding
shear stress constant (Figure 17a, as in Figure 11) creates a
vertical slice parallel to the relative supply axis. Similarly,
varying transport stage while holding supply constant,
either by increasing slope (Figure 17b, as in Figure 13) or
by decreasing grain size (Figure 17c, as in Figure 15),
creates a vertical slice through the erosion function surface
that bends away from the thresholds of motion and alluviation, and toward the threshold of suspension and maximum
bedrock exposure. The variable transport stage slices
(Figures 17b and 17c) bend because transport stage occurs
on both axes, as sediment transport capacity, in the denominator of the relative supply axis, depends on transport stage
in equation (8).
[53] The inverse relationship between bedrock incision
rate and excess shear stress at high-transport stages
(equation (24a) and Figures 12– 17), which results from
the tendency of saltation trajectories to grow more elongated
with increasing excess shear stress (equation (16) and
Figure 9), is an unanticipated feature of the saltationabrasion model. It suggests that there is a fundamental limit
to how quickly bedrock can be eroded by the process of bed
load abrasion. The critical shear stress at which erosion rate
reaches a maximum for variable excess shear stress corresponds to a critical relative sediment supply rate, which for
low excess shear stress is approximately Qs/Qt = 0.25. This
can be seen most clearly by considering the derivative of
equation (24b) (the dimensional form of the saltationabrasion model without the suspension term) with respect to
3=2
k1 D3=2
qt
s g
¼ :
4k2
4
ð30Þ
The critical relative sediment supply of 0.25 results from
the ratio of the exponents on excess shear stress in
equation (24b) (i.e., 0.5/2.0). The expression for the
derivative of equation (24a), the saltation-abrasion model
including the suspension term, with respect to excess shear
stress, is considerably more complex. The critical relative
supply is no longer a constant, but becomes an increasing
function of excess shear stress.
[54] Figure 18 is a contour map of the derivative of the
nondimensional erosion rate (E*) predicted by the full
saltation-abrasion model (equation (24a)), with respect to
transport stage, showing how the critical relative sediment
supply varies between 1.0 at the threshold of suspension to
0.25 at the threshold of motion. The point (t*/t*c, Qs/Qt)
where the curves of erosion rate versus excess shear stress
(Figures 13, 15, and 17) cross the curve of dE*/d(t*/t*c) = 0
(Figure 18) corresponds to the maximum possible erosion
Figure 17. Dimensional model results plotted in nondimensional setting to illustrate that curves plotted in Figures 11,
13, and 15 are, in effect, slices though the nondimensional
erosion function surface. (a) Curves from Figure 11, variable
sediment supply. (b) Curves from Figure 13, variable slope.
(c) Curves from Figure 15, variable grain size.
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For the stream power model, assuming wear primarily by
sediment impacts,
ef ¼ ksp ev ¼
ksp 2Y
kv s2T
ð34Þ
and for the saltation abrasion model (equation (24a))
qs
ef ¼ 0:08Rb g
tb uw
2 !3=2
1=2 t*
qs
u*
1
1
:
1
qt
tc*
wf
ð35Þ
Figure 18. Contour lines of constant derivative of the
nondimensional erosion rate (E*) with respect to transport
stage, as a function of transport stage and relative sediment
supply. Sensitivity of bedrock incision rate to changes in
transport stage is greatest near the thresholds of motion and
alluviation and declines with increasing transport stage.
Region where derivative becomes negative is indicated for
the full version of the model (equation (24a)) and for when
the saltation hop length extrapolation to account for the
transition to suspension is omitted (equation (24b)).
rate for the given fixed input values. Figure 18 also shows
that erosion rate is most sensitive to changes in excess shear
stress near the threshold of alluviation (Qs/Qt = 1).
4.5. Implications for the Stream Power Model
[55] The nondimensional framework is useful for understanding the differences between the saltation-abrasion model
and other models for river incision into bedrock incision. Here
we briefly consider the widely used model [e.g., Whipple and
Tucker, 1999] in which incision rate is assumed to be
proportional to stream power per unit bed area (w)
E ¼ ksp w ¼ ksp rw gQw S ffi ksp tb uw
Note that for the saltation-abrasion model, unlike the stream
power model, stream power erosional efficiency is independent of rock strength; energy is dissipated by fluid
friction and grain impacts in the same proportions whether
the rocks are strong or weak. Equations (34) and (35) can be
used to explicitly evaluate the roles of sediment supply and
grain size in controlling the stream power efficiency of the
abrasion process, and in influencing the value of ksp.
[57] Figure 19 shows contours of equal stream power
erosional efficiency as a function of relative sediment supply
and transport stage, assuming that all erosion takes place by
saltating bed load impacts. The efficiency surface peaks as
expected at a relative sediment supply of 0.5, where the tools
and cover effects are balanced. However, the efficiency peak
occurs at t*/t*c = 2.7, a low transport stage relative to the
nondimensional erosion rate peak of t*/t*c = 12 (Figure 16).
This suggests that moderate magnitude-frequency discharges are the most efficient events, in terms of energy
dissipation, for eroding bedrock by bed load abrasion.
[58] Note that the common assumption that Ksp is
constant for a given rock type [e.g., Stock and Montgomery,
1999] is equivalent to assuming that efficiency should plot
in this parameter space as horizontal plane. Note also that
the peak efficiency is less than 1%, indicating that nearly all
stream energy is lost to heat by turbulent and viscous fluid
friction while only a negligible fraction of total stream
ð31Þ
to illustrate how the saltation-abrasion model can be used to
disaggregate the many factors lumped into the dimensional
erosional efficiency coefficient ksp. Several studies have
attempted to calibrate ksp at a landscape scale [e.g., Stock
and Montgomery, 1999], assuming that ksp primarily
represents the role of rock resistance. However, many other
factors are implicitly represented by ksp that influence the
efficiency of the erosion process for a given rate of energy
expenditure, most notably sediment supply and grain size.
[56] The power per unit bed area expended by the river in
eroding rock (we) can be expressed as the product of erosion
rate and the impact energy required to erode a unit volume
of rock
we ¼ Eev :
ð32Þ
We can define a dimensionless stream power erosional
efficiency (ef) as the ratio of we to the total unit power
expended by the flow (w)
ef ¼
we
:
w
ð33Þ
Figure 19. Contours of nondimensional efficiency of
stream power in bedrock wear, as a function of transport
stage and relative sediment supply (equation (35)). Peak
efficiency is less than 1% and occurs at relatively low
transport stage.
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Table 4a. Kv Calibration From Wear by Dropped Particle
Experiments
st, MPa
rs, kg/m3
Mp, kg
Me, kg
ei, J
Nd
1.12
1.12
1.12
3.44
3.44
3.44
2300
2300
2300
2300
2300
2300
0.074
0.074
0.072
0.084
0.087
0.087
0.0101
0.0087
0.0138
0.0012
0.0009
0.0012
0.722
0.729
0.706
0.824
0.853
0.853
100
100
100
100
100
100
kv
1.32
1.54
9.38
1.33
1.84
1.38
1012
1012
1011
1012
1012
1012
power contributes to rock wear. This result is independent
of grain size as well as rock strength.
5. Comparison With Experimental Data
[59] In this section we compare erosion rates predicted by
the saltation-abrasion model with laboratory measurements
of bedrock wear by saltating bed load in rotational bedrock
abrasion mills [Sklar and Dietrich, 2001]. The primary goal
of this comparison is calibrating the one free parameter in
the model (kv), the dimensionless constant in equation (4)
relating the energy required to erode a unit volume of rock
(ev) to the elastic strain energy of the rock at failure (sT/2Y).
We also use the experimental data to evaluate two key
assumptions: (1) that the threshold energy for detachment
(et) can be neglected and (2) that the abrasion rate of the
bedrock bed becomes negligible as the transport mode shifts
from bed load to suspension.
[60] The data from the bedrock abrasion mills are not
appropriate, however, for fully testing the saltation-abrasion
model because the experimental conditions are different
from those assumed for the model in two important respects.
First, the rotational geometry of flow in the abrasion mills
produces a strong secondary flow, in which water descends
along the walls and rises in the center of the vortex, that
results in significant cross stream and vertical components of
the stress field at the bedrock bed. In contrast, the model
derivation above only considered the downstream stress
component (tb). The secondary circulation sweeps sediment
to the center of the bedrock disks where wedge-shaped
deposits form and expand outward with increasing sediment
loading [Sklar and Dietrich, 2001, Figure 4b]. Second,
unlike a reach of river channel, the rotational abrasion mill
is a closed system, so that partial bed coverage results from
a partitioning of sediment additions into mobile and
immobile fractions, rather than from a channel adjustment
to maintain steady state between sediment input and output.
As a result, partial bed coverage cannot be directly related to
sediment flux, as the model assumes, but rather depends on
the mass of sediment present in the abrasion mill.
[61] The bedrock abrasion mill data most useful for
calibrating kv are from runs with very low sediment loading,
such as where a single 70 gram particle was used in testing
the variation in erosion rate with rock strength, with
bedrock and sediment composed of the same lithology
[Sklar and Dietrich, 2001, Figure 3]. In this case, the
differences in the pattern of partial bed coverage between
the model and abrasion mills do not affect the results
because no significant partial coverage occurred. Similarly,
the low sediment loading for runs exploring the variation in
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erosion rate with grain size, for constant sediment mass
[Sklar and Dietrich, 2001, Figure 5], can be used to
consider the effects of including the threshold of suspension
and neglecting the threshold of detachment in the saltationabrasion model.
5.1. Calibrating the Rock Strength Parameter (kv)
[62] Use of the abrasion mill data to constrain the value of
kv requires estimates of the rate and energy of grain impacts,
which we estimate from observations of sediment motion
made through clear-walled abrasion mills and from the
predictions of the saltation-abrasion model.
[63] Because of the uncertainty in these estimates we first
conducted simple impact wear tests in which we eroded
bedrock disks by repeatedly dropping particles from a
known height (1.0 m), in order to know the impact energy
and number of impacts precisely. The impact energy of a
dropped particle ei = MpgHd where Mp is the mass of the
impacting particle and Hd is the height of drop. From
equation (4) we obtain an expression for kv for the case of
dropped particles
kv ¼ 2ei Nd rs Y =Me s2T
ð36Þ
where Nd is the total number of drops (100 in each case), Me
is the total mass of bedrock eroded, and rs is the density of
bedrock. Table 4a lists the conditions and results for drop
tests on two artificial sandstones; both tests indicate an
order of magnitude estimate of kv 1 1012.
[64] To apply the saltation-abrasion model to the experimental abrasion mill setting, we use observations of
sediment motion to estimate qs and tb, and rearrange
equation (24a) to solve for kv for the case of a single
abrasive grain
0:5
2 !0:5
0:08Rb gY qs t*
u*
1
kv ¼
1
wf
E tc*
s2T
ð37Þ
neglecting the cover term (1 Qs/Qt) because there are no
other sediment grains for the single abrasive grain to
interfere with. Assuming sediment motion can be represented by travel along a circular path at the mean radial
distance of 0.06 m, a single grain covers a circumferential
distance (Lc) of 0.38 m per revolution. With the propeller
Table 4b. Kv Calibration From Single-Grain Abrasion Mill
Experiments
st, MPa rs, kg/m3 E, g/hr
Rock Type
b
Artificial (20:1)
Artificial (6:1)
Artificial (4:1)
Sandstone
Graywacke
Limestone
Welded tuff
Quartzite
Andesite
0.163
0.448
1.12
1.583
9.1
9.78
10.9
19
24.4
2300
2300
2300
2450
2500
2600
2600
2600
2600
215
71
5.1
2.9
0.22
0.21
0.036
0.008
0.030
E,a m/s
7
8.69 10
2.87 107
2.06 108
1.11 108
8.18 1010
7.51 1010
1.26 1010
2.86 1011
1.06 1010
kv
4.07
1.63
3.63
3.37
1.38
1.30
6.23
9.09
1.47
1012
1012
1012
1012
1012
1012
1012
1012
1012
Erosion rate units conversion made with bedrock disc area of 0.03 m2.
Numbers in parenthesis indicate sand to Portland cement ratios for
artificial sandstone.
a
b
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(Table 4b), roughly consistent with an order of magnitude
estimate of kv = 1 1012.
Figure 20. Comparison of saltation-abrasion model predictions with wear rate measurements from bedrock
abrasion mill experiments [Sklar and Dietrich, 2001], as a
function of grain size, with nondimensional rock strength
parameter kv = 3 1012. Thick solid and dashed lines
correspond to equations (24a) and (24b), respectively. Thin
lines show deviation of model predictions from measured
data for values of the detachment threshold energy (et)
increasing from zero by factors of 10.
speed set to 1000 rpm, we observed 70 gram particles
rotating with an average angular velocity of 70 rpm, so that
average ‘‘downstream’’ sediment velocity us = 0.44 m/s.
Total sediment flux Qs = Mpus/Lc = 0.081 kg/s, and, for a
total radial width (rw) of 0.075 m, sediment flux per unit
width qs = Qs/rw = 1.08 kg/m s.
[65] A 70 gram abrasive grain has a nominal diameter of
0.038 m. We observed that grains larger than this size would
either not move at all, or would only move sporadically.
Grains with a diameter of about 0.045 m appeared to be
closest to the threshold of motion, vibrating in place and
occasionally sliding or rolling a short distance. Assuming a
low value of t*c = 0.03 to account for the low friction angle
for a single grain on a nearly planar bedrock surface
[Buffington and Montgomery, 1997], and rs = 2600 kg/m3,
the ‘‘downstream’’ component of the average boundary
shear stress tb = Ds(rs rw)gt*c = 22 Pa. For the 0.038 m
diameter grain, transport stage t*/t*c = 1.2 and the ratio u*/
wf = 0.19. At this transport stage, the model prediction of
downstream sediment velocity us = 0.49 m/s agrees well
with the observed average sediment velocity. The model
prediction of impact rate Ir = 90 impacts/m2 s is also
consistent, for the bedrock disc area of 0.03 m2, with the
observed frequency of 2 to 3 audible grain impacts per
second.
[66] To solve for kv using equation (37), we use the
measured wear rates for the set of runs in which bedrock of
various strengths was eroded by a single abrasive grain
composed of the same rock type [Sklar and Dietrich, 2001,
Figure 3], assuming as before a constant elastic modulus Y =
5 104 MPa [Clark, 1966]. Calculated values of kv average
3 1012, and range between 1 1012 and 9 1012
5.2. Testing Assumptions Regarding Erosion
Threshold (Et) and Hop Length (Ls)
[67] We next use the abrasion mill data to test the assumptions of a negligible threshold of erosion (et = 0) and infinite
saltation hop lengths above the threshold of suspension
(equation (14)). Data from runs in which grain size was
varied, holding total sediment mass and all other variables
constant, are most appropriate, because no partial coverage
occurred at the low sediment loading of 70 g. In the variable
grain size runs limestone bedrock was not abraded by
limestone gravel, but rather by quartzite gravel (0.01 m <
Ds < 0.035 m), vein quartz-rich mixed pebbles (0.003 m <
Ds < 0.01 m), and quartz sand (Ds < 0.002 m). Because
these sediments are stronger than the limestone bedrock, we
use kv = 3 1012 to account for the experimentally observed
threefold increase in erosion rates when quartzite gravels
were used instead of gravel composed of the same rock
type as the bedrock [Sklar and Dietrich, 2001, Figure 3].
[68] Figure 20 shows that the erosion rates predicted by
the saltation-abrasion model (equation (24a)) closely follow
the observed variation in bedrock wear as grain size
decreases from near the threshold of motion (Ds 0.035 m) to the threshold of suspension (Ds 0.001 m).
Note that in contrast to Figures 14 and 15, we plot erosion
rate on a logarithmic scale in Figure 20 so that the low
erosion rates for sand-sized grains can be distinguished
from zero. Also plotted in Figure 20 are curves for model
predictions using nonzero values of the erosion threshold
(et), and for the case of et = 0 but without the infinite hop
length suspension term (i.e., equation (24b)). As et is
increased from zero by successive orders of magnitude the
model predictions increasingly deviate from the measured
values. A nonzero erosion energy threshold affects the
predicted erosion rates most strongly for the smaller grain
sizes, because impact energy depends on the fourth power
of grain size (equation (21)).
[69] Erosion rates for sand sizes are greatly overpredicted
by the version of the model that lacks the suspension term
(i.e., equation (24b)). Neither version of the model adequately predicts erosion rates for medium sand (Ds 0.0005 m), because the model does not represent the
dynamics of suspended sediment transport in which
sediment is exchanged between a bed load layer and the
water column. In the rotational abrasion mills, sand grains
are carried down to and across the bed by the secondary
circulation, resulting in some bedrock wear. Note that no
wear was measured for fine sand (Ds 0.0002 m) despite
the occurrence of some sand transport across the bed.
6. Discussion
[70] The saltation-abrasion model is intended to represent
the dynamics of bed load abrasion for the full range of
sediment transport and supply conditions. Therefore all
channels incising into bedrock predominantly by bed load
abrasion should plot somewhere on the nondimensional
erosion function surface depicted in Figure 16. For example,
the ‘‘reference’’ reach of the South Fork Eel River shown in
Figure 1b, which is mostly mantled with coarse alluvium,
should plot near the upper left corner of Figure 16, not far
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above the threshold of motion during incision events, with
the bed mostly buried. In contrast, the reach of the Oregon
Smith River shown in Figure 1a, which receives little coarse
sediment supply, should plot nearer to the lower right
corner, with high bedrock exposure and large excess shear
stress during incision events. Because the saltation-abrasion
model was developed at the timescale of individual discharge events, the dynamic variation in channel conditions
and incision rate due to changes in discharge and sediment
supply can be represented by shifts in position on the
erosion function surface.
[71] To apply the model at much longer, geomorphically
significant timescales, we need to address tradeoffs between
magnitude and frequency inherent in the distribution of
discharges and sediment supply events that an incising
channel might experience. The most frequent discharges
typically do not exceed the threshold of motion shear stress
and thus move no bed load and cause no bedrock wear.
Peak erosion rates predicted by the saltation-abrasion model
occur at large excess shear stresses, which in most incising
channels are likely to occur only during rare extreme
discharges. In our selection of ‘‘reference site’’ conditions
(Table 3) we assume for simplicity that the integrated effect
of the full range of discharge and sediment supply events
can be represented by a dominant discharge that moves all
the annual coarse sediment supply and a fraction of the total
runoff in a small fraction of the total time. In alluvial
channels with wide adjacent floodplains, the geomorphically dominant events have moderate magnitude and recurrence intervals of 1 – 2 years [Wolman and Miller, 1960].
Baker and Kale [1998] have argued that the magnitude of
the dominant discharge in bedrock channels should scale
with the rock resistance, with more resistant rocks being
eroded primarily by infrequent extreme events.
[72] An important implication of our model is that extreme
magnitude events may not be the most effective in eroding
bedrock, instead, more frequent moderate magnitude events
may be responsible for the majority of bedrock wear. This is
because of the reduced frequency of particle impact with
increasing shear stress, the same effect that imposes an
ultimate limit to the rate of erosion by saltating bed load
(Figure 16). In addition, the coupling between discharge and
sediment supply will strongly influence which discharges are
most important in eroding bedrock. Some extreme events,
like the 1964 storm of record on Redwood Creek and other
N. California Coast Range rivers, result in meters of aggradation and complete burial of underlying bedrock [Lisle,
1982]. Large magnitude events can also strip away alluvial
cover, or mobilize coarse alluvial armor, and erode bedrock
that may have been buried for many years. Whether big
storms are net importers or exporters of sediment may
primarily depend on how much time has passed since the
last big event, and thus on how much sediment has been
produced by hillslope processes [Howard, 1998].
[73] An inherent limitation of the mechanistic approach
adopted here is that we have only considered a single
erosional process and thus probably underpredict total
incision rate, particularly at high excess shear stresses.
However, many of the effects of sediment predicted by
the saltation-abrasion model influence the efficiency of
other incision mechanisms as well. For example, partial
bed cover by transient alluvial deposits will prevent incision
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by all other erosional mechanisms, with the exception of
dissolution. Impacts by saltating bed load will certainly
influence and in some lithologies may control the rate of
plucking by enlarging and extending fractures along planes
of weakness [Whipple et al., 2000a]. Increased fracture
density due to multiple bed load impacts exposes fresh
mineral surfaces to weathering and dissolution, enhancing
rates of chemical weathering and dissolution. Bed load
abrasion may also promote cavitation by creating small pits
and other irregularities in the bedrock surface that favor
flow separation and bubble formation at high-flow velocities [Barnes, 1956; Whipple et al., 2000a].
[74] By focusing only on abrasion by bed load and
assuming that the shift to suspended transport is equivalent to grains taking saltation hops of infinite length
(equation (14)), we do not account for the possibility of
bed lowering by suspended load abrasion. Abrasion by
suspended load has been argued to dominate wear of
boulders and bedrock outcrops protruding high into the
flow [Hancock et al., 1998; Whipple et al., 2000a] and the
walls of narrow bedrock slots [Whipple et al., 2001].
However, the long-term rate of channel incision will be
controlled by the erosion rate of the lowest portions of the
channel bed, where bed load will be concentrated. The only
proposed theory for bedrock erosion by suspended load
[Whipple et al., 2000a, 2000b] assumes, by analogy with
vertical aeolian wear profiles [Anderson and Haff, 1988],
that sediment concentration, and thus particle impact rate,
scales with mean downstream fluid velocity. However, as
Whipple et al. [2000a] acknowledge, this holds only when
an unlimited supply of suspendable sediment is available on
the bed, as occurs in lowland sand bedded channels but not
in incising bedrock channels where suspended sediment
concentration is controlled by supply from upstream.
[75] Nonetheless, grains small enough to travel in suspension may contribute to lowering of bedrock channel beds.
Turbulent bursts and sweeps, and flow separation over bed
roughness elements will bring suspended grains into contact
with exposed bedrock. Moreover, contrary to our assumption
of infinite hop lengths for suspended sediment, some amount
of suspendable sediment will remain as bed load in the nearbed boundary layer due to the exchange of sediment between
bed load and suspended load that underlies conventional
suspended sediment transport theory [e.g., McLean, 1992].
Although neglecting abrasion by fine sediments transported
predominantly in suspension may result in underpredicting
abrasion rates at very high excess shear stresses, the low
erosional efficiency of fine sediments seen in bedrock
abrasion mill experiments [Sklar and Dietrich, 2001]
suggests that the extent of underprediction is small.
[76] The saltation-abrasion model could be improved by
treating channel roughness as a dynamic variable. In all the
calculations reported here we kept the roughness parameter
constant for simplicity. A more realistic approach would
account for the effect of form drag in reducing the shear
stress available for sediment transport and bed load abrasion.
The influence of grain size on skin drag should also be
included, as well as the roughness effect of bed load sediment
motion. Few data are available to help guide a mechanistic
parameterization of bedrock channel roughness. In channels
with approximately planar beds and rectangular cross sections, form drag should be at a minimum when there are few
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alluvial deposits due to low relative sediment supply; form
drag would grow with increasing bed coverage. Other
channels with highly irregular bedrock topography may be
most rough at low coverage, so that form drag may reach a
minimum at high bed coverage.
[77] Our assumption of a planar bedrock bed neglects the
possible feedbacks between differential bedrock wear and
the process of partial bed alluviation. At low relative
sediment supply, planar beds may be unstable because
bed load may preferentially move through local low points,
increasing the impact rate and thus the erosion rate in the
depression at the expense of neighboring high points. This
positive feedback would grow until the form drag associated
with the bed depressions reduced the shear stress available
to transport bed load enough to force sediment deposition to
occur. Bed load impacts would then preferentially erode the
exposed higher points, limiting the growth in the small-scale
relief of the bedrock bed. Temporal fluctuations in relative
sediment supply would have a similar effect, shifting the
focus of bed load wear between higher and lower portions
of the bedrock bed. Topographic high points that protrude
well into the flow, above the reach of bed load abrasion, will
be eroded primarily by abrasion by suspended sediments
[Whipple et al., 2000a]. Irregular bed topography will also
influence the saltation impact angle, and thus the fraction of
total kinetic energy that contributes to deformation wear.
Thus bedrock high points that do not protrude above the
zone of bed load transport will erode preferentially on their
upstream faces, leading to downstream migration of rock
bed forms [e.g., Tinkler and Parrish, 1998].
[78] Another important simplification in the saltationabrasion model is our assumption that bed cover is linearly
proportional to excess transport capacity. In addition to
neglecting the threshold of alluviation, we also do not
account for the interactions between bed sediments and
the water surface at critical flow. In flume experiments with
Froude number 1, we have observed the sudden formation
of antidunes as sediment supply and partial bed coverage
reach a threshold value [Sklar and Dietrich, 2002]. The
large increase in form drag associated with the growth of
antidunes decreases the transport capacity resulting in rapid
bed alluviation without further increases in the rate of
sediment supply. In a separate set of experiments [Sklar and
Dietrich, 2001], we have also observed bedrock wear under
a thin but complete alluvial cover, presumably because
sufficient impact energy from mobile grains striking
stationary grains is transmitted to the underlying bedrock.
We expect that a more sophisticated treatment of the
controls on partial bed cover and its insulating effects might
alter the shape of the nondimensional erosion function
surface (Figure 16), but would not alter the fundamental
constraints that the cover and tools effects impose on rates
of bedrock erosion.
[79] Perhaps the most important simplifying assumption
in the saltation-abrasion model is our use of a uniform grain
size to represent the wide distribution of grain sizes supplied
to incising channels. Grain size has a profound effect on
predicted incision rates, through (1) the threshold of motion,
(2) by scaling the effect of shear stress (i.e., nondimensional
transport stage), and (3) in influencing the height of the
erosion function surface (Figure 16). The grain size distribution of sediment supplied to the channel network will
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strongly influence the extent of alluvial cover and the
availability of bed load abrasive tools. As Howard [1980]
has shown for coarse alluvial channels, the critical threshold
of motion is set by coarsest grain size fraction supplied in
concentrations sufficient to dominate channel spanning
deposits. The grain size distribution will largely determine
the fraction of the total load that moves as bed load and thus
affects the magnitude of the sediment supply variable (Qs).
[80] Little is known about what controls the size distribution of sediments supplied to channels by adjacent hillslopes. The initial grain size distribution is likely to depend
on the bedrock lithology and its tectonic history, the
climatically influenced weathering rate and style, and hillslope sediment transport processes. Once sediments reach
the channel network, the distribution evolves rapidly due to
selective transport [Paola et al., 1992], lateral sorting
[Paola and Seal, 1995], and particle breakdown in transport
[Kodama, 1994]. Downstream fining can be interrupted by
resupply of coarse sediments at junctions with steep loworder tributaries [Rice, 1998].
7. Conclusion
[81] We have developed a mechanistic model for bedrock
incision by saltating bed load using well established empirical relationships for impact wear of brittle materials and the
motion of saltating bed load grains. The model predicts that
maximum incision rates occur at moderate sediment supply
rates, relative to sediment transport capacity, because of a
tradeoff between the availability of abrasive tools and the
partial alluviation of the bed. Incision rates also peak at
intermediate levels of excess shear stress due to a tradeoff
between particle impact energy and impact frequency. The
predicted bed load abrasion rate can be expressed in a
compact nondimensional form as a function of excess shear
stress and sediment supply relative to sediment transport
capacity. We nondimensionalize erosion rate by grain size,
and by rock resistance, which scales with the square of rock
tensile strength. In this parameter space, the saltationabrasion model defines a unique functional surface bounded
by four essential constraints on incision rate imposed by
sediment supply, grain size and the dynamics of sediment
motion. The rate of bed load abrasion will be zero at shear
stresses below the threshold of motion and above the threshold of suspension, and when the supply of coarse sediment is
either negligible or exceeds the sediment transport capacity.
[82] The saltation-abrasion model illustrates the benefits of
building geomorphic process models from the detailed
mechanics of individual erosional mechanisms. Directly
coupling bed load transport with bedrock wear by saltating
grain impacts leads to several important insights: (1) grain size
is a key control on bedrock incision rate because the threshold
of motion introduces a minimum shear stress required to erode
bedrock, and because the erosional efficiency of a given shear
stress depends on the grain size; (2) incision rate is most
sensitive to changes in shear stress when sediment supply is
nearly equal to sediment transport capacity because small
increases in shear stress can produce large changes in the
extent of bedrock exposure (Figure 12); and (3) there is an
upper limit to the rate at which bed load abrasion can erode
bedrock due to the reduced frequency of bed load impacts
with increasing shear stress (Figure 16), which suggests that
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SKLAR AND DIETRICH: BEDROCK INCISION BY SALTATING BED LOAD
other erosional mechanisms may be dominant under toolspoor or high excess shear stress conditions.
[83] Several priorities for future research emerge from
this work, including understanding: (1) the controls on the
grain size distribution supplied to river channels; (2) how
coarse sediment is transported through bedrock bedded
channels; (3) the role of the grain size distribution and the
bedrock bed roughness in influencing the formation and
growth of alluvial deposits; and (4) how to integrate incision
processes other than abrasion by bed load into a comprehensive mechanistic model for fluvial erosion of bedrock.
Notation
Ds grain diameter (m).
ef dimensionless stream power erosional
efficiency.
E bedrock erosion rate (m/s).
E* nondimensional erosion rate per unit rock
resistance.
Fe fraction of bed exposed.
Fr Froude number.
g gravitational acceleration (m/s2).
Hd particle drop height (m).
Hs saltation hop height (m).
Hw flow depth (m).
ksp stream power erosional efficiency (m/s W).
kv rock resistance coefficient.
Ir impact rate per unit area per unit time (1/m2 s).
Ls saltation hop length (m).
Lsd hop length, descending limb (m).
Lsu hop length, rising limb (m).
Me mass of rock eroded (kg).
Mp particle mass (kg).
Ms total sediment mass supplied to abrasion mills
(kg).
Nd number of particle drops.
n mannings roughness coefficient.
qs sediment supply per unit width (kg/ms).
Qs total sediment supply (kg/s).
qt sediment transport capacity per unit width
(kg/m s).
Qt total sediment transport capacity (kg/s).
Qw discharge (m3/s).
Rb nondimensional bouyant density.
Rep particle Reynolds number.
rh hydraulic radius (m).
rw total radial width of abrasion mill disc (m).
S channel slope.
td saltation descent time (s).
Ui particle impact velocity (m/s).
Ur particle rebound velocity (m/s).
us horizontal sediment velocity (m/s).
usi Horizontal sediment velocity on impact (m/s).
uw mean flow velocity (m/s).
u* flow shear velocity (m/s).
Vi volume eroded per impact.
W channel width (m).
wf particle fall velocity (m/s).
wsd mean saltation descent velocity (m/s).
wsi vertical sediment velocity on impact (m/s).
Y Young’s modulus of elasticity (MPa).
a saltation impact angle (degrees).
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b elastic strain energy (J/m3).
DeN net impact energy transferred (J).
DeD impact deformation energy (J).
DeL impact energy lost (J).
ei impact energy of dropped particle (J).
et detachment threshold energy (J).
ev unit volume detachment energy (J/m3).
g nondimensional excess shear stress [t*/
t*c 1].
m kinematic viscosity (kg/ms).
rs sediment density (kg/m3).
rw water density (kg/m3).
sT rock tensile strength (MPa).
tb mean boundary shear stress (Pa).
t* nondimensional shear stress.
t*c nondimensional critical shear stress.
t*/t*
c transport stage.
)
(t*/t*
c susp transport stage at threshold of suspension.
w total unit stream power (W).
we unit stream power expended in rock wear (W).
[84] Acknowledgments. We thank G. Hauer, A. Howard, J. Kirchner,
A. Luers, C. Reibe, J. Rice, J. Roering, M. Stacey, and J. Stock for
insightful discussions. We are also grateful to K. Whipple and three
anonymous reviewers whose comments significantly improved the manuscript. This work was supported by the National Center for Earth-Surface
Dynamics, by NSF grant EAR 970608, and by a Switzer Environmental
Fellowship to L.S.S.
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Figure 1. Photographs of two incising rivers with contrasting alluvial cover. (a) Smith River, Oregon
coast range mountains. (b) South Fork Eel River, northern California coast range mountains.
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